In the given figure, ABCD is a parallelogram. E is a point on DC. If AE bisects $∠$ A and BE bisects $∠$ B, then which of the following statements is correct?

AB = BC

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AB = 2 BC

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AB = 12BC

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AB = 3 B

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Solution

The correct option is B
AB = 2 BC

Let $∠$ A = 2x°

$∠$ DAE = $∠$ EAB = x° …… (AE bisects $∠$ A)

ABCD is a parallelogram …… (given)

$∠$ EAB = $∠$ AED = x° (alternate angles)

ADE is isosceles (base angles are equal to x°)

Therefore, DE = AD ………. (i)

$∠$ B = 180° - 2x° ……(adjacent angles of a parallelogram)

$∠$ ABE = $∠$ EBC = 90°- x° …… (AE bisects $∠$ A)

ABCD is a parallelogram …… (given)

$∠$ EAB = $∠$ AED = 90°- x° (alternate angles)

BCEis isosceles (base angles are equal to x°)

Therefore, CE = CD ………. (ii)

But, AD = CD (opposite sides of a parallelogram are equal)

Therefore, AD = BC = DE = CE

But, DE + EC = DC = AB (opposite sides of a parallelogram)

AB = 2 BC

Hence (B)

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In the given figure, ABCD is a parallelogram. E is a point on DC. If AE bisects $∠$ A and BE bisects $∠$ B, then which of the following statements is correct?

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In the given figure, ABCD is a parallelogram. E is a point on DC. If AE bisects ∠A and BE bisects ∠B, then which of the following statements is correct?

In the given figure, ABCD is a parallelogram. E is a point on DC. If AE bisects $∠$ A and BE bisects $∠$ B, then which of the following statements is correct?